Hessian nilpotent polynomials and the Jacobian conjecture
By Wenhua Zhao. 2004
50 citations recorded by the source corpus.
Let z = (z 1 ,···,z n ) and let A = Σ n i=1 ∂ 2 ∂z 2 i be the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to what we call the vanishing conjecture: for any homogeneous polynomial P(z) of degree d = 4, if Δ m P m (z) = 0 for all m ≥ 1, then Δ m P m+1 (z) = 0 when m > > 0, or equivalently, Δ m P m+1 (z) = 0 when m > 3 2(3 n-2 -1). It is also shown in this paper that the condition Δ m P m (z) = 0 (m ≥ 1) above is equivalent to the condition that P(z) is Hessian nilpotent, i.e. the Hessian matrix Hes P(z) = (∂ 2 P ∂z i ∂z j ) is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived.
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/405a91c15ae9445f7b89796ff750fc51a17bcfc3
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In Geometry and Topology
Citations in the corpus
- Generalizations of the Image Conjecture and the Mathieu Conjecture (2009, background): “It is also the case for the Jacobian conjecture and, more generally, the vanishing conjecture [ Z1 ], [Z2] on differential operators (of any order) with constant coefficients via theirconnections with the IC (See [Z3]).”
- Images of commuting differential operators of order one with constant leading coefficients (2009, background): “The motivation for the conjecture above is the following theorem proved in [ Z2 ].”